diff --git a/ChangeLog.d/improve-doc-on-ecp-curve-optimized-representation.txt b/ChangeLog.d/improve-doc-on-ecp-curve-optimized-representation.txt
new file mode 100644
index 0000000..8fdc588
--- /dev/null
+++ b/ChangeLog.d/improve-doc-on-ecp-curve-optimized-representation.txt
@@ -0,0 +1,3 @@
+Features
+   * The documentation of mbedtls_ecp_group now describes the optimized
+     representation of A for some curves. Fixes #8045.
diff --git a/include/mbedtls/ecp.h b/include/mbedtls/ecp.h
index 0e678a3..a89d4d2 100644
--- a/include/mbedtls/ecp.h
+++ b/include/mbedtls/ecp.h
@@ -197,6 +197,27 @@
  * odd prime as mbedtls_ecp_mul() requires an odd number, and
  * mbedtls_ecdsa_sign() requires that it is prime for blinding purposes.
  *
+ * The default implementation only initializes \p A without setting it to the
+ * authentic value for curves with <code>A = -3</code>(SECP256R1, etc), in which
+ * case you need to load \p A by yourself when using domain parameters directly,
+ * for example:
+ * \code
+ * mbedtls_mpi_init(&A);
+ * mbedtls_ecp_group_init(&grp);
+ * CHECK_RETURN(mbedtls_ecp_group_load(&grp, grp_id));
+ * if (mbedtls_ecp_group_a_is_minus_3(&grp)) {
+ *     CHECK_RETURN(mbedtls_mpi_sub_int(&A, &grp.P, 3));
+ * } else {
+ *     CHECK_RETURN(mbedtls_mpi_copy(&A, &grp.A));
+ * }
+ *
+ * do_something_with_a(&A);
+ *
+ * cleanup:
+ * mbedtls_mpi_free(&A);
+ * mbedtls_ecp_group_free(&grp);
+ * \endcode
+ *
  * For Montgomery curves, we do not store \p A, but <code>(A + 2) / 4</code>,
  * which is the quantity used in the formulas. Additionally, \p nbits is
  * not the size of \p N but the required size for private keys.
@@ -223,8 +244,11 @@
 typedef struct mbedtls_ecp_group {
     mbedtls_ecp_group_id id;    /*!< An internal group identifier. */
     mbedtls_mpi P;              /*!< The prime modulus of the base field. */
-    mbedtls_mpi A;              /*!< For Short Weierstrass: \p A in the equation. For
-                                     Montgomery curves: <code>(A + 2) / 4</code>. */
+    mbedtls_mpi A;              /*!< For Short Weierstrass: \p A in the equation. Note that
+                                     \p A is not set to the authentic value in some cases.
+                                     Refer to detailed description of ::mbedtls_ecp_group if
+                                     using domain parameters in the structure.
+                                     For Montgomery curves: <code>(A + 2) / 4</code>. */
     mbedtls_mpi B;              /*!< For Short Weierstrass: \p B in the equation.
                                      For Montgomery curves: unused. */
     mbedtls_ecp_point G;        /*!< The generator of the subgroup used. */
@@ -992,6 +1016,26 @@
 
 #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED)
 /**
+ * \brief           This function checks if domain parameter A of the curve is
+ *                  \c -3.
+ *
+ * \note            This function is only defined for short Weierstrass curves.
+ *                  It may not be included in builds without any short
+ *                  Weierstrass curve.
+ *
+ * \param grp       The ECP group to use.
+ *                  This must be initialized and have group parameters
+ *                  set, for example through mbedtls_ecp_group_load().
+ *
+ * \return          \c 1 if <code>A = -3</code>.
+ * \return          \c 0 Otherwise.
+ */
+static inline int mbedtls_ecp_group_a_is_minus_3(const mbedtls_ecp_group *grp)
+{
+    return grp->A.MBEDTLS_PRIVATE(p) == NULL;
+}
+
+/**
  * \brief           This function performs multiplication and addition of two
  *                  points by integers: \p R = \p m * \p P + \p n * \p Q
  *
diff --git a/library/bignum.c b/library/bignum.c
index f2a8641..f02b1ac 100644
--- a/library/bignum.c
+++ b/library/bignum.c
@@ -258,6 +258,10 @@
     return (mbedtls_mpi_uint) 0 - (mbedtls_mpi_uint) z;
 }
 
+/* Convert x to a sign, i.e. to 1, if x is positive, or -1, if x is negative.
+ * This looks awkward but generates smaller code than (x < 0 ? -1 : 1) */
+#define TO_SIGN(x) ((((mbedtls_mpi_uint) x) >> (biL - 1)) * -2 + 1)
+
 /*
  * Set value from integer
  */
@@ -270,7 +274,7 @@
     memset(X->p, 0, X->n * ciL);
 
     X->p[0] = mpi_sint_abs(z);
-    X->s    = (z < 0) ? -1 : 1;
+    X->s    = TO_SIGN(z);
 
 cleanup:
 
@@ -326,16 +330,35 @@
  */
 size_t mbedtls_mpi_lsb(const mbedtls_mpi *X)
 {
-    size_t i, j, count = 0;
+    size_t i;
     MBEDTLS_INTERNAL_VALIDATE_RET(X != NULL, 0);
 
+#if defined(__has_builtin)
+#if (MBEDTLS_MPI_UINT_MAX == UINT_MAX) && __has_builtin(__builtin_ctz)
+    #define mbedtls_mpi_uint_ctz __builtin_ctz
+#elif (MBEDTLS_MPI_UINT_MAX == ULONG_MAX) && __has_builtin(__builtin_ctzl)
+    #define mbedtls_mpi_uint_ctz __builtin_ctzl
+#elif (MBEDTLS_MPI_UINT_MAX == ULLONG_MAX) && __has_builtin(__builtin_ctzll)
+    #define mbedtls_mpi_uint_ctz __builtin_ctzll
+#endif
+#endif
+
+#if defined(mbedtls_mpi_uint_ctz)
     for (i = 0; i < X->n; i++) {
-        for (j = 0; j < biL; j++, count++) {
+        if (X->p[i] != 0) {
+            return i * biL + mbedtls_mpi_uint_ctz(X->p[i]);
+        }
+    }
+#else
+    size_t count = 0;
+    for (i = 0; i < X->n; i++) {
+        for (size_t j = 0; j < biL; j++, count++) {
             if (((X->p[i] >> j) & 1) != 0) {
                 return count;
             }
         }
     }
+#endif
 
     return 0;
 }
@@ -796,9 +819,8 @@
         }
     }
 
-    if (i == 0 && j == 0) {
-        return 0;
-    }
+    /* If i == j == 0, i.e. abs(X) == abs(Y),
+     * we end up returning 0 at the end of the function. */
 
     if (i > j) {
         return 1;
@@ -880,7 +902,7 @@
     MPI_VALIDATE_RET(X != NULL);
 
     *p  = mpi_sint_abs(z);
-    Y.s = (z < 0) ? -1 : 1;
+    Y.s = TO_SIGN(z);
     Y.n = 1;
     Y.p = p;
 
@@ -1068,7 +1090,7 @@
     MPI_VALIDATE_RET(A != NULL);
 
     p[0] = mpi_sint_abs(b);
-    B.s = (b < 0) ? -1 : 1;
+    B.s = TO_SIGN(b);
     B.n = 1;
     B.p = p;
 
@@ -1086,7 +1108,7 @@
     MPI_VALIDATE_RET(A != NULL);
 
     p[0] = mpi_sint_abs(b);
-    B.s = (b < 0) ? -1 : 1;
+    B.s = TO_SIGN(b);
     B.n = 1;
     B.p = p;
 
@@ -1436,7 +1458,7 @@
     MPI_VALIDATE_RET(A != NULL);
 
     p[0] = mpi_sint_abs(b);
-    B.s = (b < 0) ? -1 : 1;
+    B.s = TO_SIGN(b);
     B.n = 1;
     B.p = p;
 
diff --git a/library/bignum_core.c b/library/bignum_core.c
index 8bf819c..ae0b94a 100644
--- a/library/bignum_core.c
+++ b/library/bignum_core.c
@@ -686,16 +686,16 @@
 
 static size_t exp_mod_get_window_size(size_t Ebits)
 {
-    size_t wsize = (Ebits > 671) ? 6 : (Ebits > 239) ? 5 :
-                   (Ebits >  79) ? 4 : 1;
-
-#if (MBEDTLS_MPI_WINDOW_SIZE < 6)
-    if (wsize > MBEDTLS_MPI_WINDOW_SIZE) {
-        wsize = MBEDTLS_MPI_WINDOW_SIZE;
-    }
+#if MBEDTLS_MPI_WINDOW_SIZE >= 6
+    return (Ebits > 671) ? 6 : (Ebits > 239) ? 5 : (Ebits >  79) ? 4 : 1;
+#elif MBEDTLS_MPI_WINDOW_SIZE == 5
+    return (Ebits > 239) ? 5 : (Ebits >  79) ? 4 : 1;
+#elif MBEDTLS_MPI_WINDOW_SIZE > 1
+    return (Ebits >  79) ? MBEDTLS_MPI_WINDOW_SIZE : 1;
+#else
+    (void) Ebits;
+    return 1;
 #endif
-
-    return wsize;
 }
 
 size_t mbedtls_mpi_core_exp_mod_working_limbs(size_t AN_limbs, size_t E_limbs)
diff --git a/library/debug.c b/library/debug.c
index 56bc3f6..0983cb0 100644
--- a/library/debug.c
+++ b/library/debug.c
@@ -144,7 +144,6 @@
 
     debug_send_line(ssl, level, file, line, str);
 
-    idx = 0;
     memset(txt, 0, sizeof(txt));
     for (i = 0; i < len; i++) {
         if (i >= 4096) {
@@ -202,17 +201,54 @@
 }
 #endif /* MBEDTLS_ECP_LIGHT */
 
-#if defined(MBEDTLS_BIGNUM_C)
 #if defined(MBEDTLS_PK_USE_PSA_EC_DATA)
+static void mbedtls_debug_print_ec_coord(const mbedtls_ssl_context *ssl, int level,
+                                         const char *file, int line, const char *text,
+                                         const unsigned char *buf, size_t len)
+{
+    char str[DEBUG_BUF_SIZE];
+    size_t i, idx = 0;
+
+    mbedtls_snprintf(str + idx, sizeof(str) - idx, "value of '%s' (%u bits) is:\n",
+                     text, (unsigned int) len * 8);
+
+    debug_send_line(ssl, level, file, line, str);
+
+    for (i = 0; i < len; i++) {
+        if (i >= 4096) {
+            break;
+        }
+
+        if (i % 16 == 0) {
+            if (i > 0) {
+                mbedtls_snprintf(str + idx, sizeof(str) - idx, "\n");
+                debug_send_line(ssl, level, file, line, str);
+
+                idx = 0;
+            }
+        }
+
+        idx += mbedtls_snprintf(str + idx, sizeof(str) - idx, " %02x",
+                                (unsigned int) buf[i]);
+    }
+
+    if (len > 0) {
+        for (/* i = i */; i % 16 != 0; i++) {
+            idx += mbedtls_snprintf(str + idx, sizeof(str) - idx, "   ");
+        }
+
+        mbedtls_snprintf(str + idx, sizeof(str) - idx, "\n");
+        debug_send_line(ssl, level, file, line, str);
+    }
+}
+
 void mbedtls_debug_print_psa_ec(const mbedtls_ssl_context *ssl, int level,
                                 const char *file, int line,
                                 const char *text, const mbedtls_pk_context *pk)
 {
     char str[DEBUG_BUF_SIZE];
-    mbedtls_mpi mpi;
-    const uint8_t *mpi_start;
-    size_t mpi_len;
-    int ret;
+    const uint8_t *coord_start;
+    size_t coord_len;
 
     if (NULL == ssl              ||
         NULL == ssl->conf        ||
@@ -223,32 +259,21 @@
 
     /* For the description of pk->pk_raw content please refer to the description
      * psa_export_public_key() function. */
-    mpi_len = (pk->pub_raw_len - 1)/2;
+    coord_len = (pk->pub_raw_len - 1)/2;
 
     /* X coordinate */
-    mbedtls_mpi_init(&mpi);
-    mpi_start = pk->pub_raw + 1;
-    ret = mbedtls_mpi_read_binary(&mpi, mpi_start, mpi_len);
-    if (ret != 0) {
-        return;
-    }
+    coord_start = pk->pub_raw + 1;
     mbedtls_snprintf(str, sizeof(str), "%s(X)", text);
-    mbedtls_debug_print_mpi(ssl, level, file, line, str, &mpi);
-    mbedtls_mpi_free(&mpi);
+    mbedtls_debug_print_ec_coord(ssl, level, file, line, str, coord_start, coord_len);
 
     /* Y coordinate */
-    mbedtls_mpi_init(&mpi);
-    mpi_start = mpi_start + mpi_len;
-    ret = mbedtls_mpi_read_binary(&mpi, mpi_start, mpi_len);
-    if (ret != 0) {
-        return;
-    }
+    coord_start = coord_start + coord_len;
     mbedtls_snprintf(str, sizeof(str), "%s(Y)", text);
-    mbedtls_debug_print_mpi(ssl, level, file, line, str, &mpi);
-    mbedtls_mpi_free(&mpi);
+    mbedtls_debug_print_ec_coord(ssl, level, file, line, str, coord_start, coord_len);
 }
 #endif /* MBEDTLS_PK_USE_PSA_EC_DATA */
 
+#if defined(MBEDTLS_BIGNUM_C)
 void mbedtls_debug_print_mpi(const mbedtls_ssl_context *ssl, int level,
                              const char *file, int line,
                              const char *text, const mbedtls_mpi *X)
@@ -324,19 +349,21 @@
         mbedtls_snprintf(name, sizeof(name), "%s%s", text, items[i].name);
         name[sizeof(name) - 1] = '\0';
 
+#if defined(MBEDTLS_RSA_C)
         if (items[i].type == MBEDTLS_PK_DEBUG_MPI) {
             mbedtls_debug_print_mpi(ssl, level, file, line, name, items[i].value);
         } else
+#endif /* MBEDTLS_RSA_C */
 #if defined(MBEDTLS_ECP_LIGHT)
         if (items[i].type == MBEDTLS_PK_DEBUG_ECP) {
             mbedtls_debug_print_ecp(ssl, level, file, line, name, items[i].value);
         } else
-#endif
+#endif /* MBEDTLS_ECP_LIGHT */
 #if defined(MBEDTLS_PK_USE_PSA_EC_DATA)
         if (items[i].type == MBEDTLS_PK_DEBUG_PSA_EC) {
             mbedtls_debug_print_psa_ec(ssl, level, file, line, name, items[i].value);
         } else
-#endif
+#endif /* MBEDTLS_PK_USE_PSA_EC_DATA */
         { debug_send_line(ssl, level, file, line,
                           "should not happen\n"); }
     }
diff --git a/library/ecp.c b/library/ecp.c
index 870773c..f9b6672 100644
--- a/library/ecp.c
+++ b/library/ecp.c
@@ -1255,7 +1255,7 @@
     MPI_ECP_SQR(rhs, X);
 
     /* Special case for A = -3 */
-    if (grp->A.p == NULL) {
+    if (mbedtls_ecp_group_a_is_minus_3(grp)) {
         MPI_ECP_SUB_INT(rhs, rhs, 3);
     } else {
         MPI_ECP_ADD(rhs, rhs, &grp->A);
@@ -1526,7 +1526,7 @@
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
 
     /* Special case for A = -3 */
-    if (grp->A.p == NULL) {
+    if (mbedtls_ecp_group_a_is_minus_3(grp)) {
         /* tmp[0] <- M = 3(X + Z^2)(X - Z^2) */
         MPI_ECP_SQR(&tmp[1],  &P->Z);
         MPI_ECP_ADD(&tmp[2],  &P->X,  &tmp[1]);